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Doctoral Dissertations
University of Connecticut Graduate School
9-4-2013
A Mechanochemical Study into the Growth and
Shape Maintenance of Rod-shaped Bacteria
Akeisha Belgrave
[email protected]
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A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria
Akeisha Belgrave, PhD
University of Connecticut, 2013
Rod-shaped bacteria grow in length without changing their width [1, 2]. A major feature
of bacterial cell growth is the remodeling of the cell wall, the primary structure giving the
bacterium its shape and structural integrity. In many rod-shaped bacteria, the process by which
the cell wall gets larger as the cell grows involves severing bonds that link the existing cell wall
material together in order to insert and bind new material. Coordination between severing and
insertion is likely necessary in order to prevent the cell wall from rupturing due to the large
pressure difference between the inside and outside of the cell. As more bacteria are becoming
resistant to antibiotics, which typically target cell wall synthesis, understanding the regulatory
mechanisms involved in bacterial growth and morphology may provide new paradigms for
treating infections. Although there have been many hypotheses proposed about the mechanisms
underlying the regulation of cell width, the actual process by which bacteria grow and maintain
their shape remains unclear [3, 4].
To begin to address this question, we developed a simple mechanochemical model
explaining the dependence of cell length and degree of crosslinking on the replication rate of
rod-shaped bacteria [5]. In this model we observed that faster growing cells are less crosslinked
and longer in length than slower growing cells, as has been experimentally measured. Since our
model predicts that faster growing cells have weaker cell walls, a prediction of the model is that
the fractional change in length of a cell upon osmotic shock should be greater for faster growing
Akeisha Belgrave – University of Connecticut, 2013
cells. We tested this prediction and found good agreement between the model and experiments.
Finally, in order to understand the consequences of crosslinking fraction and glycan strand length
on the elastic properties of the cell wall, we developed a 2D percolation model of the
peptidoglycan meshwork. By simulating the effects of a given stress on this network, we were
able to define a constitutive relationship for the cell wall as a function of crosslink fraction and
glycan strand length. This model allows us to observe how mechanical and biochemical
properties can affect cell shape.
A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria
Akeisha Belgrave
B.S., Norfolk State University, 2008
A Dissertation
Submitted in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
at the
University of Connecticut
2013
Copyright by
Akeisha Belgrave
2013
APPROVAL PAGE
Doctor of Philosophy Dissertation
A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria
Presented by
Akeisha Belgrave, B.S.
Major Advisor _________________________________________________________
Dr. Ann Cowan
Associate Advisor ______________________________________________________
Dr. Charles Wolgemuth
Associate Advisor ______________________________________________________
Dr. Ji Yu
Associate Advisor ______________________________________________________
Dr. William Mohler
Associate Advisor ______________________________________________________
Dr. Michael Blinov
University of Connecticut
2013
Acknowledgements
I would like to send great thanks to the wonderful people in my life that made my work a
fighting success. I cannot say this journey was an easy one, but it was a life-changing and
successful one.
To Dr. Ann Cowan and Dr. Raquell Holmes, thank you both for all the assistance you gave to me
from the start of the program to get me adjusted and educated in this area of studies. Coming in
to a biomedical program with a background in Physics and Mathematics I wondered how I was
going to catch up with the rest of the crowd understanding the concepts and terms needed to
progress throughout this journey. I must say, being from another country, with no family here,
you two felt like extended mothers, and I felt like I could come to you with anything whether it
was related to work or just my life.
To my co-workers and lab members, Eunji, Erika, Olena, Jing, Sofya, thank you guys for the fun
times I needed to get a break when things got too stressful. Mike you are a life saver, thank you
for your kind assistance in my experiments, thank you Dhruv (Mr. Happy Hour). Thank you for
being there to assist when I needed you. Elnaz, Jing and Pilhwa thank you for helping me with
any problems I encountered in my mathematical calculations. Thanks to Dr. Koen Visscher for
assisting me in the final stages of my work with resources unavailable to me at the time.
To my family, especially my mom, my sister and my Auntie Lynda, thank you for your support,
you guys were always there to listen and motivate me when I felt like giving up. Special thanks
to my close friends Anicia, Shari, DeAnn, Tamara, Vanessa, Sharon and Ryan for the support
throughout this entire journey. You were always there when I needed you even from more than a
thousand miles away.
Now for the big thank you, to my advisor Dr. Charles Wolgemuth; the most down to earth,
EXTREMELY smart person in my life through it all. Thank you for being so helpful and patient
with me. Even when I would look at you with blank expression, you always found a way to
explain the things in the best way. You were fun to work with, and your ability to know almost
everything truly inspired me. I could not have learnt more in my life than from working under
your guidance. This would not have been possible without you. Thank you.
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Dedication
This work is dedicated to my mother Adorial Maxwell-Hazell and most importantly to our
heavenly father. Mom you fulfill your dreams even when times got tough and never gave up.
You are truly a person of strength and inspiration. Thank you for your continued motivation and
assistance when times seemed too difficult.
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Table of Contents
Acknowledgments ……………………………………………………………………… i
Dedication ……………………………………………………………………………….ii
List of figures …………………………………………………………………………… iv
Chapter 1: Introduction
1
1.1 General Bacterial Physiology …………………………………………………………2 - 3
1.2 Bacterial Cell Wall ……………………………………………………………………3 - 6
1.3 Bacterial Synthesis …………………………………………………………………....6 - 7
1.4 Bacterial Biosynthetic pathway ……………………………………………………..........7
1.5 Peptidoglycan Synthetic machinery ………………………………………….............8 -11
1.6 Bacterial Growth …………………………………………………………………….11 - 12
Chapter 2: Elasticity and biochemistry of growth relate replication rate to cell length and
crosslink density in rod-shaped bacteria (Biophysical Journal)
2.1 Introduction……………………………………………………………………………….13
2.2 A Mechanochemical Model of Bacterial Morphology .............................................. 14 - 19
2.3 Results and Discussion ……………………………………………………………..20 – 25
Chapter 3: The relationship between replication rate and cell wall strain in Escherichia coli
suggests mechanisms of cell wall synthesis
3.1 Introduction ………………………………………………………………………….26 - 28
3.2 Methods ……………………………………………………………………………...29 - 30
3.3 Results and Discussion……………………………………………………………....30 - 36
Chapter 4: The effects of peptidoglycan structure on the elasticity of the Gram-negative cell wall
4.1 Introduction ………………………………………………………………………....37 - 39
4.2 Modeling the Gram-negative cell wall………………………………………………39 - 43
4.3 Results and Discussion ……………………………………………………………...44 - 47
Chapter 5: Conclusions and suggestions for future research ……………………………....48 - 51
Bibliography …………………………………………………………………….………….52-59
iii
List of Figures
1.1. Schematic representation of the structure of the bacterial cell wall in Gram-negative cells
(e.g. E. coli) and Gram-positive cells (e.g. B. subtilis) …………………………………….11
1.2. Schematic representation of a rod-shaped bacterium and the bacterial PG network ….......12
1.3. Schematic representation of Penicillin binding protein (PBP) grouping. HMWs, LMWs, and
-lactamases. The HMW group subdivided into Class A and Class B…………………….13
2.1 Schematic of bacterial growth .……………………………………………………………..25
2.2 Mean cell length increases with division rate consistent with experiments on E. coli while
crosslinking decreases .................................................................................................................26
3.1. Schematic representation of the flow of water molecules due to changes in extracellular
solute …………………………………………………………………………………………....35
3.2 The replication rate of E. coli MG1655 increases approximately linearly with temperature
and the average cell length increases with the replication rate, whereas, the width shows a slight
decrease ……………………………………………………………………………………..….36
3.3 Representative images of E. coli MG1655 cells before and after the addition of 500 mM
sucrose and the fractional change in length and width increase with replication rate ..…..……37
4.1 Peptidoglycan percolation model for a gram-negative bacterium at 100% connectivity (i.e., Pp =
Pg = 1), and 50% crosslink density (Pp = 0.5) and 96% mean glycan chain length (Pg = 0.96), within
the range appropriate for E. coli …………………………………………………………………... 48
4.2 Elastic moduli of a single layer of peptidoglycan at varying crosslink densities and glycan
chain length ………………………………………………………………………………….....49
iv
Chapter 1
Introduction
Bacteria are known to be the cause of many common illnesses (i.e. stomach ulcers, strep throat
and many other bacterial infections). In general, these infections are treated using a range of
antibiotics; some common ones being penicillin, tetracycline, streptomycin and azithromycin.
Most antibiotics work by interfering with the composition of or the formation of the bacteria’s
cell wall. Therefore, pharmaceutical companies have made this structure the major drug target to
treat bacterial infections. The increasing prevalence of antibiotic-resistant bacteria requires new
treatment methods and/or new locations for drug targeting. In order to increase the efficacy of
these drugs, a more precise understanding of the structure and synthesis of the bacterial cell wall
is needed.
The bacterial cell wall, which is made of a meshed material known as peptidoglycan (PG), is the
shape determining factor in bacteria. This elastic material protects the cell against mechanical
stresses applied to the cell from the environment and also resists the large internal turgor
pressure of the cell, which can exceed atmospheres of pressures and can easily rupture the cell
membrane. Bacterial cells differ in size, shape, cellular and genetic makeup. The most common
shapes are cocci (spherical), bacillus (straight-rod), or vibrio (curved rod) shaped. Not only do
bacteria possess various shapes, but these shapes are maintained throughout many generations.
The simple genetic makeup and reproducibility of shape and size throughout generations also
make these organisms an ideal model for understanding growth and form in biology.
1
1.1
General Bacterial Physiology
Bacteria have a fairly simple basic structure. The inside of the cell is known as the cytoplasm
and contains water, proteins, and DNA. The cytoplasm is separated from the outside world by
the inner (or cytoplasmic) membrane, a thin bi-layer membrane that is largely impermeable.
Channels and pumps in the membrane control the flow of ions, nutrients and water into and out
of the cell. The cell wall lies just outside the inner membrane. In addition to these three
structures, some bacterial species also have a second bilayer membrane, the outer membrane that
encloses the cell wall.
Bacteria are often generically classified as being either Gram positive or Gram negative. This
classification is based on a technique that uses a dye that turns the cells purple. After washing,
however, only Gram-positive cells retain the dye, while Gram-negative cells fade to pink, thus
giving them the corresponding names. Gram staining detects the peptidoglycan in the cell wall
of bacteria [6]. Gram positive bacteria stain because they do not have an outer membrane and
because they typically have thicker cell walls than Gram negatives [7].The two model organisms
for these types of bacteria are Escherichia coli (Gram negative) and Bacillus subtilis (Gram
positive). These two types of bacteria are categorized as bacillus due to their elongated rodshape. On average, E. coli is about 1.0 m wide and 2.0 – 4.0 m long [8, 9]. B. subtilis is about
0.8 m wide by 4.0 m long [10]. Non-pathogenic E.coli lives in the digestive tracts of humans
and animals. The feces from these animals can then contaminate foods and water, leading to
bacterial infections. B. subtilis are non-pathogenic bacteria that are normally found in soil and
vegetation. These bacteria are known to contaminate food but seldom cause food poisoning. The
manageable size of the genome and proteome of these bacteria and the availability of mutants
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with altered morphologies make these two bacteria excellent choices for understanding cell
shape at the molecular level and will be used throughout this work.
1.2
Bacterial Cell wall
The primary component of the bacterial cell wall is a macromolecular structure known as the
peptidoglycan (PG), which makes up 5-90% of the cell wall mass in bacteria [11]. As
mentioned, this structure is located outside the cytoplasmic membrane. The PG is composed of
glycan strands, which are polymeric chains of sugars and amino acids that are linked together via
covalent glycosidic and peptide bonds [12]. Because these bonds are strong, the PG is a cohesive
mechanical structure which maintains cell shape. The PG even retains the same shape and
approximately the same size when it is isolated from the rest of the bacterium [13, 14, 15, 16, 17,
18, 19]. The peptidoglycan is inflated by the internal osmotic pressure of the cell and acts to
protect the cell against externally-applied forces, thereby reducing the likelihood of cell lysis
[20]. The internal osmotic gauge pressure (or turgor pressure) is on the order of hundreds of kilo
pascals (kPa) to a few MPas, with Gram positive bacteria typically having larger turgor pressures
(15 to 20 atm) than Gram negative (0.8 to 5 atm) respectively [21-24]. These large turgor
pressures require cells to have strong cell walls to counteract this pressure [25].
The thickness of the cell wall under Gram staining defines whether a cell is Gram-positive or
Gram-negative. The percentage of PG is higher in Gram-positive than in Gram-negative. In
Gram-negative bacteria, the cell wall is a thin elastic monolayer (i.e. 1-9 nm) located within the
periplasm, a viscous or gel-like media that makes up 40% of the cell volume and contains some
of the important cellular enzymes[26, 27, 28,29,30] (Figure 1.1A). On the other hand, in Gram3
positive bacteria, the cell wall is a thicker, multi-layered structure (i.e. 30-100 nm) interwoven
with teichoic acids and other proteins [30] (Figure 1.1B). Unlike-, Gram-negative bacteria, these
bacteria do not possess an external membrane. The intact cell wall of Gram-negative bacteria can
withstand pressures up to 5 atm and Gram-positives can withstand pressures up to 50 atm [2123].
A molecular view of the PG layer shows that it is a meshwork of polysaccharide (glycan) strands
comprised of N-acetylmuramic acid (MurNAc) and N-acetylglucosamine (GlcNAc) monomers
interconnected by peptide stems [16-32]. These glycan strands are then connected together by
peptide stems of five D- and L-amino acids, including diaminopimelic acid that is attach to a
lactyl group within the MurNAc monomer. The peptide stems are wrapped helically around the
glycan strand in such a way that subsequent stems on a given strand are roughly perpendicular to
each other [13, 19]. In Gram-positive bacteria there may exist an additional peptide bridge that
connects the two peptide stems. The glycan strands are considered to be relatively stiff
compared to the more flexible peptides [19, 33]. Despite some controversy, the glycan strands
are generally thought to be circumferentially oriented about the cell [14].
Various methods have been used to measure glycan strand length and the degree of
polymerization of the strands in the cell wall. The most widely used method is reversed-phase
HPLC, which separates the products obtained after complete degradation of the PG with
muramidases that degrade the glycan strands or amidases that cleave the attached peptides from
the glycans [27, 34]. Exponentially growing, wild-type E.coli cells have a mean glycan length
(mGCL) of 25-35 disaccharide units, but the glycan strands can be up to 80 disaccharide units in
4
length [12, 34, 27, 35]. In B. subtilis a mean glycan length (mGCL) of 1,300 disaccharides was
found [36].
The peptide stems are of a fixed length but not all of these stems are crosslinked to neighboring
glycan strands. The degree of crosslinking of E.coli has been measured to be around 20% [31].
Due to the location of the peptide stems on the glycan strands, the number of possible peptide
stems is a function of glycan chain length (GCL). Lower degrees of crosslinking presumably
lead to weaker cell walls. Therefore, it is expected that the Young’s modulus (i.e. a measure of
the stiffness of an elastic material) of the cell wall will be lower in cells with lower degrees of
crosslinking.
1.3 Bacterial Synthesis
Imagine yourself inflating a cylindrically-shaped balloon. The balloon will have an appreciable
pressure filling the latex structure causing it to take shape. Now imagine that, by some strange
means, chunks of the latex material from random points along the balloon were cleaved while
maintaining constant internal pressure. These chunks are then replaced by larger pieces of latex.
One would likely expect this mechanism to produce a balloon that is both longer and wider.
However, the method just described is a fairly good analogy for the processes that occur within
the cell wall of a growing rod-shaped bacterium. Bacterial growth requires synthases to attach
new PG material to the existing PG. In concert with these synthases, hydrolases work to cleave
the existing PG to allow insertion of new PG units. Therefore, bacterial biosynthesis involves (1)
5
the severing of existing PG material; (2) insertion of newly synthesized PG units; and (3) recrosslinking of peptide stems.
1.4 Bacterial Biosynthetic pathway
Synthesis of the cell wall begins in the cytoplasm with the synthesis of UDP-activated precursors
of the MurNAc-pentapeptide and GlcNAc monomers. The UDPMurNAc-pentapeptide is then
anchored to a membrane carrier molecule undecaprenylphosphate, also known as bacteprenol.
This complex forms the Lipid I precursor. The attachment of GlcNAc to the Lipid I precursor at
the inner side of the cytoplasmic membrane forms a subunit (lipid II) that is then flipped across
the inner membrane [12, 31].
During cell wall turnover, autolysins (which are bacterial enzymes) are directed to the site of PG
digestion and growth to sever both the glycosidic bonds as well as the pentapeptide bridges [12].
The carrier bacteprenol is then released from Lipid II, and the breaks in the existing PG layer are
filled by insertion of the two remaining monomers by glycosidase enzymes. Transglycosidase
enzymes then catalyze the formation of new glycosidic bonds between the new disaccharide unit
and the existing PG layer. New glycan strands in the PG are then crosslinked through the
formation of peptide bridges by transpeptidases, leading to a strong cell wall. Studies suggest
that many rod-shaped bacteria (including E. coli and B. subtilis) elongate by inserting material
along the lateral cell wall with little to no insertion at the poles [16-20, 37].
6
1.5
Peptidoglycan Synthetic machinery
In order to maintain the integrity of the cell, cleavage and insertion of material in the cell wall
must be well coordinated. Although internal pressure plays a role in cell shape by inflating the
cell wall, the detailed shape of the bacterium is regulated by specific proteins within the
periplasm that coordinate their behavior to influence cell shape. As mentioned in the previous
section, hydrolases are required to cleave existing PG material while synthases are required to
attach the new PG material to the existing cell wall material. In this section I will identify the
known key regulatory proteins and their function during bacterial cell wall synthesis.
The main PG-synthesizing enzymes, the penicillin-binding proteins, or PBPs, involved in the last
stages of PG synthesis have been identified [8, 38-39, 40-42]. PBPs can be either high molecular
weight (HMW) or low molecular weight (LMW) or -lactamases [43]. HMW PBPs can be
further divided into class A or B enzymes made up of either monofunctional or multifunctional
PBPs (Figure 1.3) [43, 44].
The HMW PBPs, PBP2 and PBP3 are monofunctional Class B enzymes found in E.coli. These
proteins are anchored to the cytoplasmic membrane and are known to be responsible for the
insertion of newly synthesized PG precursors into the cell wall. Localization studies using
fluorescent microscopy identified newly incorporated PG precursors in live cells and showed that
these enzymes are involved in PG synthesis [45]. Apart from the previously mentioned
monofunctional enzymes PBP2 and PBP3, there also exist enzymes with bifunctional properties.
Some bifunctional enzymes with both transglycosylase and transpeptidase activity are PBP1A,
PBP1B, and PBP1C. It was shown in [46] that deletion of either of the HMW class A enzymes
7
PBP1A or PBP1B in E.coli cells was lethal to the cell and hence both of them are required for
growth.
Along with the PBPs, specific hydrolases are required to cleave covalent bonds to allow for
insertion of new precursors. Cell wall synthesis requires the cooperation of glycosidases that
work to polymerize the precursors that form the glycan strands. This process is followed by
crosslinkage of these strands through the formation of peptide bridges by transpeptidases. The Cterminus of the HMW PBPs is the penicillin-binding domain, which is responsible for
crosslinking the PG layer, while the N-terminus has transglycosylase activity making it
important in the elongation of glycan strands (transglycosylation) and the formation of peptide
crosslinks (transpeptidation) [1].
The bacterial cytoskeleton plays an important role in directing bacterial cell wall synthesis. The
integral membrane proteins RodA and FtsW have a high degree of similarity to each other and
are involved in the translocation of the lipid-linked PG precursors [47, 48]. These membrane
proteins have been shown to interact with the transpeptidases PBP2 and PBP3 [49] and may be
responsible for directing the PG precursors to the transpeptidases for insertion into the wall.
RodA, FtsW and the actin-like rod-shape determining protein MreB have been shown to be
required for cell elongation by being part of the pathway for insertion of new PG units along the
lateral walls of in E. coli [40] and B. subtilis [50]. The absence of these proteins in rod-shaped
bacteria cells leads to spherical cells. MreB was first believed to be organized as a helix located
beneath the cytoplasmic membrane that spanned the length of the cell [50]; however, recent
experiments have shown that this is an artifact [51]. MreC and MreD reside outside the inner
membrane and depletion of these two proteins in rod-shaped cells also leads to a spherical shape.
In E. coli [52], MreC interacts with both MreB and MreD forming a multiprotein complex. The
8
depletion of the multiprotein complex MreBCD highlights its importance in maintaining cell
width since depletion led to increased cell width [52].
1.6
Bacterial Growth
The work presented here describes research into the mechanisms underlying the synthesis of the
bacterial cell wall that occurs when bacteria are actively replicating. The changes in length
and/or size of the bacterium during this process could be defined as growth. However, it is
important to note that in the literature bacterial growth often refers to the process of replication.
That is, the bacterial growth rate often denotes the time rate of change of the number of bacterial
cells, as opposed to the change in size of an individual bacterium. Throughout this dissertation, I
will try to avoid this ambiguity by referring to the change in the number of bacteria as the
replication rate and the time rates of change of length and width, etc. will be specifically stated
as such.
In regard to bacterial replication, it is also important to note that there are four phases of growth:
the lag phase, the log phase, the stationary phase, and the death phase. The lag phase occurs just
after a bacterial culture is diluted. This is a transient period during which the culture does not
grow in size. Following this period, the cells begin to replicate and divide, with a fairly constant
time between one replication and the next. Therefore, the number of cells in the population
increases exponentially with time. During this phase, if the concentration of cells is plotted as a
function of time on a semi-log plot, a straight line is obtained. Hence the name “log phase”. The
9
slope of this line is the replication rate, which can vary depending on the strain of bacteria and
other environmental factors of the growth medium and can range from minutes up to days.
During the stationary phase, the size of the population remains constant even though some cells
start to divide, while some die. The death phase which follows the stationary phase occurs when
the death of the cells exceeds the formation of new cells.
Previous research suggests that cell morphology is correlated with the rate that cells replicate and
divide. It is known that cells that replicate faster have longer average cell lengths [53, 54-56].
Concurrently, as cells replicate faster there is a decrease in the degree of crosslinking [5]. As
previously mentioned, it is commonly believed that these synthetic reactions (i.e. severing of
peptide stems and/or glycan strands, insertion of new disaccharides and re-crosslinking of the
peptide stems) occur simultaneously due to the efficient functioning of a multi-enzyme complex.
Throughout our research we attempted to explain this dependence on the replication rate by
developing a simple mechanochemical model for the mechanics and biochemistry of PG
synthesis. In the next chapter we explain our model in full detail by including results published
in a recent paper [5] in the Biophysical Journal. This model predicts that the strain in the cell
wall should increase with replication rate. Chapter 3 describes the experiments that test this
prediction. In Chapter 4, we develop a computational model that predicts how glycan length and
the degree of peptide crosslinkage affect the mechanical properties of the PG layer in a 2D
mechanochemical model. Lastly, in Chapter 5 we conclude with a summary of this work and
suggestions for future directions.
10
Figures
(A)
(B)
Figure. 1.1: Schematic representation of the structure of the bacterial cell wall in (A) Gram-negative cells
(e.g. E. coli) that have a single layer (5-7 nm) of PG within the periplasmic space which lies between the
cytoplasmic and outer membrane, and (B) Gram-positive cells (e.g. B. subtilis) that have a multi-layered
(20-25 nm) of PG on the external side of the cytoplasmic membrane.
11
Figure 1.2: Schematic representation of a rod-shaped bacterium and the bacterial PG network. The
network consists of a repeated chain of MurNAc-pentapeptide monomers and GlcNAc monomers
forming a disaccharide unit. These chains are then crosslinked by the pentapeptide stems, thereby forming
the PG.
12
High-molecularweight (HMW)
PBPs
Class A: HMW
Low-molecularweight (LMW)
PBPs
-lactamases
Class B: HMW
Mono or
Multifunctional
PBPs
Figure 1.3. Penicillin binding proteins (PBPs) are involved in the last stages of PG synthesis. These
proteins are classified into three separate groups, HMWs, LMWs, and -lactamases. The HMW group is
subdivided into Class A and Class B.
13
Chapter 2
Elasticity and biochemistry of growth relate replication rate
to cell length and crosslink density in rod-shaped bacteria
(Biophysical Journal, Volume 104, Issue 12, 2607-2611, 18 June 2013)
2.1 Introduction
In this chapter we describe a simple, yet quantitative, model of the synthesis of the cell wall of
rod-shaped bacteria that combines the fundamental physics and biochemical kinetics involved in
this process. Our principle hypothesis is that the length of a bacterium is defined by the total
amount of material in the cell wall and how much that material is stretched by the turgor
pressure. Since the amount of material in the cell wall is due to the rate that new material is
inserted in and removed from the cell wall, the biochemical kinetics of synthesis dictates the
amount of material in the wall; however, these synthetic reactions can depend on the mechanical
state of the cell wall. We, therefore, propose a mechanochemical model that explains the
dependence of cell length and crosslinking on the replication rate. Our model shows good
agreement with existing experimental data and provides further evidence that cell wall synthesis
is mediated by multi-enzyme complexes; however, our results suggest that these synthesis
complexes only mediate glycan insertion and crosslink severing, while re-crosslinking is
performed independently.
14
2.2 A Mechanochemical Model of Bacterial Morphology
As previously described, the bacterial cell wall is made of a covalently linked, elastic material
known as peptidoglycan (PG), which is a network of glycan strands crosslinked by flexible
pentapeptide stems [3,4,5]. In order to insert new material into the PG, existing material must be
cleaved [20]. Therefore, synthesis of the PG involves (a) severing of existing peptide bonds
and/or glycan strands; (b) insertion of new dissacharide units; and (c) re-crosslinking by
formation of new peptide bonds [25]. These biochemical processes are driven by enzymes, such
as the penicillin binding proteins (PBPs). Recent evidence suggests that some aspects of cell wall
synthesis are processive and that the enzymes that drive synthesis may be localized in a multienzyme complex [4]. However, how biochemistry and biophysics conspire to change the length
of the cell without changing the width remains enigmatic, even though this process has received
a fair amount of scrutiny in the last ten years [12, 14-16, 31]. For example, computational
modeling was used to investigate the effect of insertion and severing on the morphology of rodshaped bacteria but did not directly consider the biochemical reactions that drive PG synthesis
[12]. Another group posited that the mechano-chemistry of insertion leads to dynamics that
attempt to minimize the free energy of the cell wall without describing how this dynamics could
arise from biochemical kinetics [31]. A recent model also examined processive synthesis of cell
wall growth, but did not directly model how crosslink density, elasticity and replication rate
affect cell length [14].
While efforts to understand bacterial cell width control are ongoing, an equally puzzling
phenomenon has received less attention. Bacterial cell morphology is tightly coupled to the rate
15
that cells replicate and divide. It is well known that growth conditions that increase the bacterial
replication rate (which is inversely proportional to the doubling time) also lead to populations of
cells that have a longer average length [25, 26, 31, 32]. In other words, cells that divide faster
are longer. How does faster replication lead to longer cell length?
To address this question, we take an alternative approach to modeling bacterial morphology and
construct a simple model for the mechanics and chemistry of PG synthesis. We focus on the
fundamental physics and biochemical kinetics in order to elucidate general principles of rodshaped growth. Specifically, we address the two experimental observations that cell length
increases as the replication rate of the bacteria increases, while the fraction of peptides in the PG
that are crosslinked decreases [53, 54, 55, 56].
Bacterial cell length is a consequence of two of things, the total amount of material in the cell
wall and how much that material is stretched by the turgor pressure. The amount of PG that is in
the cell wall is controlled by the rates that new PG is inserted and old PG is removed. There is
then a net insertion rate that is equal to the insertion rate minus the removal rate. This net
insertion rate times the doubling time sets how much cell wall material a given cell has. The PG
is elastic and gets stretched by the turgor pressure. The Young’s modulus of the cell wall will
depend on how crosslinked the PG is.
To construct a mathematical model that incorporates these features, we begin by considering a
single growing Gram negative cell with a single layer of PG. While conclusive evidence on the
orientation of the glycan strands is lacking, the predominant view favors alignment of the glycan
16
strands about the circumference of the cell with the peptide chains oriented parallel with the long
axis of the cell [28, 57] (Fig. 1). As mentioned above, the cell length is a function of the amount
of material and how stretched it is. Therefore, we consider the glycan strands to be hoops about
the cell circumference with each hoop defining a cross-section (Fig. 1B). We further consider
that the cell maintains a fixed number of disaccharides, 4N, per cross-section. The total amount
of material is then proportional to the total number of cross-sections, which we denote as X. Each
disaccharide in a glycan strand has one peptide stem; however, the peptide stems rotate ~90º per
disaccharide [19]. Therefore, we assume that only 1-in-4 peptides are available to crosslink any
two cross-sections, which implies that there are only N possible crosslinking sites between each
cross-section. If the average fraction of bound crosslinks per cross-section is  and each bound
peptide chain acts like a linear elastic spring with spring constant k and rest length a, the turgor
pressure P will induce a strain between cross-sections  given by
R2 P  akN   0
(2.1)
where R is the radius of the bacterium. Here we have assumed that the elastic stress in the cell
wall equilibrates fast compared to the biochemical reactions, which is a standard assumption [59,
60]. The total length of the growing cell is then equal to the number of cross-sections times the
distance between cross-sections, L = aX(1 + ).
17
To complete our model, we need to describe the biochemical reactions that occur during cell wall
synthesis (Fig. 1). Because our simple description of the cell wall ignores glycan strand length,
we only consider the kinetics of peptide severing and crosslinking, along with insertion of new
material.
We define the rate constants of severing and crosslinking to be koff and kon,
respectively. New glycan monomers are inserted between existing cross-sections at a net rate RI
(which includes insertion and turnover). Insertion can occur in one of two ways, the new
disaccharide can be inserted either with or without crosslinking it to a neighboring strand. We,
therefore, define f to be the fraction of new monomers that are inserted and crosslinked
simultaneously. The fraction of peptide bonds then obeys the kinetic equation,
 t  koff   kon 1     fRI N
(2.2)
The physics and kinetics just described hold for the crosslinking and strain between any two
cross-sections. Insertion of new disaccharide units between existing cross-sections also leads to
an increase in the number of cross-sections. Since there are 4N disaccharides per cross-section,
the time rate of change of the cross-sections is X t  RI X 4 N .
To finish our description of a single growing cell, we need to define the local insertion rate, RI.
It is important to note that this insertion rate is the rate that new material is inserted between a
given cross-section. The cell-level insertion rate is then XRI. Bacterial cell wall synthesis
involves the production of precursor molecules and enzymes in the cytoplasm. The precursor
molecules are then transported across the inner membrane in a lipid-bound form and are then
incorporated into the existing PG structure by enzymes, such as the PBPs [12]. Transport of the
18
precursor molecules across the membrane is believed to occur at a sufficient rate to match the
precursor production rate [12]. We, therefore, expect that the transport rate should be
proportional to the concentration of precursor molecules times the surface area of the cell, and
the local insertion rate (which is the insertion rate per length) should is proportional to the flux of
precursors out of the cytoplasm, which depends on the density of the precursor molecules, the
cell circumference and the number of available insertion sites. Recent experiments and
modelling have shown that ribosome production, protein production, and the replication rate of
bacteria are tightly coupled [61]. Indeed, these results suggest that the total number of molecules
of a given protein in the cell should be proportional to et, where is the replication rate [61].
Note that our model makes no assumption on what sets the replication rate. This rate is likely
determined by the nutrient capacity of the organism in the environment [61]. The cell volume is
proportional to XN2, and the cell circumference is proportional to N. Therefore, the flux of
precursor molecules out of the cytoplasm is p0et/XN, where p0 is a constant that is proportional
to the initial concentration of the precursors. We assume that new strands can only be inserted
between cross-sections at locations that are not currently crosslinked. Therefore, the insertion
rate should be RI = K0p0et(1-)/X , where K0 is a rate constant. The elongation of the cell is
then dictated by the coupled set of equations for the number of cross-sections and the fraction of
bound crosslinks:
t
X K 0 p0e 1   

t
4N
fK p et 1   

 koff   kon 1     0 0
t
NX
(2.3)
The model described so far considered a single growing cell that does not divide. For cells that
are growing in length while replicating and dividing, the exponential rate of division can balance
19
elongation and lead to a well-defined average length of the cells in the population. This average
length can be determined by the total length defined above divided by the total number of cells,
L  L M  aX 1    M , where M = M0et is the total number of bacteria.
2.3 Results and Discussion
We begin by examining the behavior of the model for a single growing cell that does divide. It is
straightforward to show that for timescales much longer than the replication rate, the elongation
model (Eqs. 2.3) predicts that the number of cross-sections and the crosslink fraction should be
K 0 p0et 1   
X
4N
k 4f
  on
kon  koff
(2.4)
Therefore, the length of the cell grows exponentially in time as
L  a 1    X

aK 0 p0 et
R 2 P 

1    1 

4N
 akN  
(2.5)
This result is consistent with the finding that when septation is blocked (such as by inhibiting
FtsZ, the protein that initiates the cytokinetic ring) the length of rod-shaped bacteria increases
exponentially in time [62].
20
For a population of bacteria, the initial amount of precursor molecules, p0, is proportional to the
initial number of cells, M0. Therefore, our model predicts that the average length of a population
of cells is
 R 2 P 
K
L  1    1 

N
 akN  
(2.6)
where K = aK0p0/4M0. Note that the ratio p0/M0 reflects the average number of precursor
molecules per cell. Therefore, the factor K is a constant that depends on the average behavior of
the population and is not dependent on the initial conditions.
Experimental data suggests that the average cell length of E. coli and B. subtilis increases with
the division rate [54-56], while the degree of crosslinking decreases [53]. (The width also
increases, but modeling that effect is outside the scope of our simple model.) The force balance
equation (Eq. 2.1) gives that the strain in the cell wall is only a function of the turgor pressure
and the fraction of cross-links. Therefore, at constant turgor pressure, the average length of the
cell only depends on the degree of crosslinking, which can be determined by Eq. 2.4. If severing
and/or crosslinking of the peptides during cell wall synthesis are mediated by a multi-enzyme
synthesis complex, then we would expect that the rates koff and kon, respectively, would be
proportional to the insertion rate and would, therefore, depend, directly or indirectly, on the
growth rate. We, therefore, consider four scenarios: (A) severing and crosslinking are
independent of insertion (koff and kon are constant); (B) severing or (C) crosslinking is linked to
21
insertion (koff or kon, respectively, are linearly functions of ); and (D) both rates are proportional
to the division rate.
From Eqs. 2.4, 2.6, one way to get an increase in length with the division rate is if koff alone is
dependent on (i.e., Scenario B)(Fig. 2A). This naturally leads to the result that the degree of
cross-linking will decrease with the division rate (Fig. 2B). This mechanism suggests that
severing and insertion are linked together during synthesis; however, crosslinking is
independent. Basically as the cell grows faster, severing occurs more rapidly while the rate of recrosslinking remains fixed. Consequently, faster replicating cells have a lower fraction of bound
peptides, and there is a larger strain in the existing peptide crosslinks. Another mechanism that
would also work is if kon = A-B, where A and B are positive constants.
This mechanism assumes that peptide crosslinking decreases as the replication rate increases,
which suggests that peptide crosslinking does not occur simultaneously with insertion. We favor
the first mechanism, as it seems more likely that a bacterium would sever old crosslinks in order
to insert new material, rather than inserting material at locations that have not yet been
crosslinked.
In order to validate this model, we set koff = kac, where ka and c are constants, and compare
the results of Eq. 2.6 to the experimental measurements of E. coli length versus division rate
given in [54,55] (Fig. 2A). Our model depends on four parameters, a dimensionless pressure,
R2P/akN, and the ratios kon/ka, c/ka, and 4f/ka. We find good agreement between our model and
the experimental results when R2P/akN ~ 0.3, kon/ka ~ 3.5, c/ka ~ 0.5, and 4f/ka ~ 0.4. These
22
parameters suggest that  is on order between 0.7-0.8. Since our model only considers the
peptides that are parallel to the long axis of the cell (i.e., half the total peptide chains), we predict
that the degree of crosslinking is around 35-40%, which is consistent with experimental
measurements in E. coli [53]. In addition, our model predicts that the strain in the cell wall
should be approximately 50%, which is consistent with the decrease in length that is observed
upon osmotic shock [63, 64]. From Eq. 1, the effective Young’s modulus for the cell wall is E =
akN/2Rl, where l is the thickness of the PG, which is ~4.5 nm for E. coli [26, 58].
Measurements of the Young’s modulus in E. coli suggest that E ~ 25 MPa [26, 58]. Therefore,
our model predicts that the turgor pressure is P ~ 3lE/5RMPa, which falls in the range of
a number of experimentally-based estimates [58, 63].
As described, our model assumes that the number of disaccharide subunits about the
circumference of the cell is fixed. This assumption may not be entirely valid as cell width also
increases with the replication rate. However, cell width is less affected by cell growth than cell
length. Using a linear function to fit the previous data on cell length and width as a function of
replication rate that was reported in [55] suggests that L = L0(1+0.42) and W = W0(1+0.19).
Therefore, a 42% change in length only corresponds with a 19% change in cell width (which
changes by approximately 50% over the range of replication rates in the experimental data,
whereas the length changes by a factor of 2). In addition, it is not clear how much of this change
in width is due to an increase in the number of disaccharides about the circumference and how
much is due to additional strain in the cell wall due to a reduction in the PG elasticity. It is then
likely that the number of disaccharides about the cell circumference (i.e., N) does not change
significantly with the replication rate.
23
Here we have developed a simple model for bacterial cell wall synthesis that couples the
biochemistry of synthesis with the biomechanics of the cell wall. The model reproduces the
observation that the average cell length in a population increase roughly linearly with the
replication rate of the bacteria. This result is a consequence of the increase in the number of
unbound peptides. Decreasing the number of bound peptides does two things. First, it reduces
the Young’s modulus for the cell, and, therefore, the turgor pressure strains the cell wall more.
Second, it increases the number of sites where new strands can be inserted, which increases the
rate that new material is incorporated into the cell wall. In order for the model to predict an
increase in unbound peptides requires that severing and insertion are coupled and independent of
re-crosslinking. This result is not consistent with the 3-for-1 model of synthesis that suggests
that all three processes occur simultaneously [12].
24
Figures
Figure 2.1. Schematic of bacterial growth. (A) A bacterium of initial length, L0, is inflated by the turgor
pressure, P. (B) The glycan strands are assumed to form circumferential hoops about the cell that are
connected by peptide chains (thin lines) that are strained an amount ε by the turgor pressure. (C) Cell wall
synthesis includes severing and recrosslinking of peptides and insertion of new disaccharide units, which
build new cross-sections between existing ones (D). (E) Insertion of new material, therefore, leads to
lengthening of the bacterium.
25
Figure 2.2. Mean cell length only increases with division rate for Scenario B (solid red line), which
is consistent with the experiments on E. coli (Data taken from (54) (blue circles) and (55) (green
squares)). Scenarios C (dashed line) and D (dotted line) led to decreases in length with division rate.
(B) The degree of crosslinking decreases for Scenario B (solid line), but increases for the other
Scenarios. The parameter values are as given in the text.
26
Chapter 3
The relationship between replication rate and cell wall strain
in Escherichia coli suggests mechanisms of cell wall synthesis
3.1 Introduction
Bacterial cells must be able to withstand large, rapid environmental changes, such as a sudden
change in the concentrations of extracellular solutes. Changes in the external solute
concentrations, known as osmotic shocks or stresses, can cause a rapid flow of water into or out
of the cell through osmosis, which consequently causes the cell’s volume to change. In hypotonic
environments the concentration of solutes is higher inside the cell, which causes a flow of water
molecules into the cytoplasm of the cell due to the osmotic pressure difference, as shown in
(Figure 3.1A). Consequently, the cell inflates. The increase in cytoplasmic volume stretches the
cell wall and increases the hydrostatic pressure in the cell. The hydrostatic pressure difference
between the inside and outside of the cell increases and eventually becomes equal to the osmotic
pressure, at which point the flow of water into the cell ceases (Figure 3.1B). Note, that the solute
concentration inside and outside of the cell does not become equal. On the other hand, if the
external solute concentration becomes more concentrated (hypertonic), then water molecules will
move out of the cytoplasm of the cell, causing the cell to shrink (Figure 3.1C).
27
In the preceding chapter, we developed a simple model for bacterial cell wall synthesis that
explains the observation that cells that replicate faster are, on average, longer. The model
suggests that severing and insertion of new material are processes that are directly coordinated
with the replication rate of the cell; however, crosslinking of the peptide chains on newly
inserted strands occurs at a rate that is independent of the replication rate. Therefore, as the
replication rate increases, the rate that the peptide bonds are severed increases while the rate that
they are re-crosslinked remains constant. Consequently, the degree of crosslinking is predicted
to decrease. The reason that the model then predicts that faster replicating cells are longer is then
two-fold. First, in a Gram-negative bacterium that has a single layered cell wall, new glycan
strands can only be inserted in between strands that are not already cross-linked. Therefore,
fewer crosslinks provide more sites for insertion of new strands. Hence, the rate that new strands
are inserted increases. In addition, because there are fewer cross-links, the cell wall is predicted
to be weaker and, therefore, is stretched more by the internal turgor pressure. That is, the model
predicts that if the turgor pressure were to be reduced by a constant amount, cells that replicate
faster would undergo a larger fractional change in length than cells that were in conditions where
they replicate slower.
In this chapter, we test the predictions of our model by using hypertonic solutions to osmotically
shock wild-type E. coli (MG1655) cells. Our experiments show that the fractional change in
length of E. coli cells increases with the division rate as predicted by our model. We also
observe that the fractional change in width does not depend as strongly on the replication rate,
which is consistent with the assumption in our model that the number of glycan monomers about
the circumference of the cell is roughly constant. These findings, therefore, validate our model
28
for lateral cell wall synthesis in Gram-negative bacteria, which then provides a basis for
developing a deeper understanding of the mechanisms underlying bacterial morphology.
3.2 Methods
Strains and Media
Wild-type K12 strain of E. coli (MG1655) was used in all experiments. The bacteria were grown
to mid-to-late log phase (OD600 of 0.45-0.65) in LB medium (Sigma) at either 23, 33, 37 or 42°C
while shaking at 150 rpm.
Measurement of the Replication Rate
Cells grown at a given temperature were washed twice with fresh LB medium and then
resuspended to a dilution of 1:10 in 1 mL of fresh LB. The cultures were then shaken at 150 rpm
at the original temperature. Every 15 minutes for up to eight hours, 1.5 mL of the culture was
micropipetted into a cuvette and the OD600 was measured using a photospectrometer. The
replication rate was then determined from the linear region of the semi-log plot of OD600 versus
time.
Sample Preparation for Osmotic Shock Experiments
Overnight 5 mL cultures of E. coli were grown in waterbaths at the desired temperature while
shaking. Prior to imaging, these cultures were diluted 1:10 in fresh LB media and grown to midlog phase (OD600 = .45-.65). Wheat Germ Agglutinin (WGA) conjugated to Alexa-Fluor 488
29
(Invitrogen) was added to 1 mL of the culture at a final concentration of 25 g/ml, in order to
label the cell wall. The samples were incubated with the dye for 20 minutes at the growth
temperature, before being pelleted and resuspended in fresh media. We followed the method
described in [64] to construct tunnel slides by adhering coverslips to glass slides using two sided
tape. 1% PEI solution was flowed into the tunnel slides and allowed to settle, before flowing in
50 l of the WGA labeled E. coli. The slides were then incubated for 10 minutes while the cells
adhered to the PEI, and again washed with LB to remove any unbound cells.
Microscopy
The tunnel slide with WGA-Alexa 488 stained cells were imaged every 20 seconds for 20
minutes using DIC and epifluorescence on a Zeiss Axio-Observer V. At 5 minutes, 50 L of 500
mM sucrose in LB was flowed into the tunnel slide, inducing hyperosmotic conditions for the E
coli. This procedure was done for cells grown at each temperature. The image processing
software, Image J [65] was then used to measure the length and width before and after the shock.
We only measured cells that were clearly in focus before and after the shock. The average
measurements from three time points before (or after) the osmotic shock were used to measure
the initial (and final) lengths and widths.
3.3 Results and Discussion
In order to determine whether the elastic strain in the cell wall is larger for cells that replicate
faster, we needed a set of growth conditions that would produce different replication rates. The
30
previous data on the dependence of cell length as a function of replication rate showed that the
average length in a population of cells is not strongly dependent on the specifics of the growth
conditions. Over a large range of different conditions, including changes in media, temperature,
and nutrients, average cell length followed the same dependence on replication rate [54, 55].
We, therefore, chose to use growth temperature to control the replication rate, as this was easily
adjustable and does not alter the nutrient availability or solute concentrations in the media.
The replication rate of bacteria is known to depend on temperature. At low temperatures,
increasing the temperature leads to higher replication rates, whereas at high temperatures, the
replication rate decreases with increases in the temperature. That is, there is an optimal
temperature at which the growth rate of a bacterial strain is most rapid. For K12 serotype E. coli,
this optimal temperature is around 42ºC [66]. Therefore, we chose to use four temperatures
between 23ºC and 42ºC to examine the relationship between elastic strain in the cell wall and
replication rate. To begin, we determined the replication rates of E. coli strain MG1655, which
we denoted as , at 23ºC, 27ºC, 33ºC and 42ºC. We found that the replication rate increases
roughly linearly in this temperature range from 1.35 hr-1 to 1.81 hr-1 (Figure 3.2A). At 37 ºC, we
found a replication rate of 1.61 hr-1, which is comparable to what has been measured previously
[55].
Next, we measured the average length and width of cells that were grown at these temperatures.
We used DIC microscopy to examine the morphology of cells that were grown to mid-log phase
at each temperature. Consistent with previous measurements, we found that the average length
of cells in a population increases roughly linearly with the replication rate (Figure 3.2B). Fitting
this data to a line, we found a slope of 0.9 m/hr, which agrees well with the value 0f 0.97 m/hr
that was reported previously [54]. Interestingly, we found that the width decreased slightly
31
between the slowest replication rate (at 23ºC) and the rate at 37ºC. The width was approximately
constant for the three largest growth rates (Figure 3.2B). This finding is different than what has
been previously reported [67, 55]; however, the range of replication rates that we investigated
covered only the high range of what was used in the previous reports. Indeed, the few data
points from the other reports that were in the same range that we considered also show a small
decrease in width at fast replication rates [55].
Next, we osmotically shocked the cells by washing in LB medium that contained 500 mM
sucrose. Because sucrose is unable to enter the cell, the addition of sucrose in the medium
increases the external osmolarity of the medium by an amount that is roughly proportional to the
concentration of sucrose [68]. Therefore, upon the addition of sucrose medium, the cells shrink
due to the effective drop in the internal turgor pressure (Figure 3.3A-D). We also find that cells
grown at higher temperatures shrink more on average than cells grown at lower temperatures
(Figure 3.3A-D). However, since cells grown at higher temperatures are also longer (Figure
3.2B), the larger change in length with higher replication rate does not confirm the predictions of
our model.
A cell that is stretched by the turgor pressure has an unstretched length and width, which we can
denote as L0 and W0, respectively. The turgor stretches these to a final length and width Lf and
Wf. The elastic strains in the cell wall are given by (Lf - L0)/L0 and (Wf - W0)/W0. These strains
should be proportional to the turgor pressure, and inversely related to the stiffness of the cell
wall. Therefore, decreasing the turgor pressure by addition of sucrose causes the elastic strains
in the cell wall to change. If the length and width after the shock are L and W, we can define the
fractional changes in length and width as (Lf - L)/L and (Wf - W)/W, respectively. These should
32
be a measure of the total strain in the cell wall and are also expected to be inversely proportional
to the cell wall stiffness in the axial and circumferential directions, respectively.
We, therefore, measured the fractional changes in length and width of cells grown to mid-late log
phase at the four temperatures that were used to determine the replication rates and average
lengths and widths. We found that the fractional change in length increased linearly with the
replication rate, going from 0.129 ± 0.001 at 23ºC to 0.191 ± 0.001 at 42ºC (Figure 3.3E) with a
p-value of 3.1 E -12 between those two temperatures. We also measured the fractional change
in width, which also increases with replication rate; however, the fractional change in width
seems to plateau at high replication rates and is approximately a factor of two smaller than the
fractional change in length (Figure 3.3B).
Here we have shown that the elastic strain in the cell wall of E. coli is dependent on the
replication rate of the bacteria. This result was predicted by the mechanochemical model for cell
wall synthesis that we developed in Chapter 2. The primary hypotheses of this model were that
cell wall synthesis in Gram-negative bacteria requires new material to be inserted between
uncrosslinked glycan strands, and that the rate of crosslinking new strands is independent of the
replication rate, whereas insertion and severing of material are proportional to the replication
rate. Therefore, our experimental results provide evidence that these mechanisms are involved in
establishing and maintaining the morphology of Gram-negative rod-shaped bacteria. However,
the results that have been described so far only provide qualitative validation of the model
predictions. Because we find the same dependence in average cell length with replication rate as
was used to determine the parameters in our model (See Chapter 2), it is possible to use the
33
mechanochemical model to predict quantitatively what is expected for the fractional change in
length. Equations 2.1 and 2.4 can be used to write the strain in the cell wall as
R 2 P  kon  koff 

akN  kon  4 f  
(0.1)
Using the parameter values previously determined, the strain is then predicted to obey

1.2  0.3 
 3.5  0.4 
(3.1)
where  is in units of hr-1. This equation predicts that the total strain in the cell wall should vary
roughly linearly with respect to replication rate for values of  less than about 5, and for values
of  in this range, the slope of the strain versus replication rate should be approximately 0.1 hr.
Since our osmotic shock experiments do not necessarily measure the total strain in the cell wall,
but rather measure the change in length for a fixed change in turgor pressure, it is better to
compare the model prediction for the slope to the experimental data. In our experiments, the
fractional change in length varied by 0.06 while the replication rate varied by 0.46 hr-1, which
gives a slope of 0.13hr, which agrees extremely well with the model prediction. It is important to
note that no further adjustment of the model parameters was necessary to get this agreement
between the model and the data. Therefore, our experimental results provide strong evidence to
support the mechanochemical model.
Our finding that the fractional change in length increases with the replication rate contrasts a
recent report that examined the strain in the cell wall of the Gram-positive bacterium B.subtilis
using osmotic shock [69]. This previous work observed no dependence of the fractional change
34
in length with replication rate. It is likely that the difference in our results from the previous
work reflects the difference in cell wall synthesis between Gram-negative and Gram-positive
cells.
Figures
(A)
(B)
(C)
Figure 3.1. (A) In a hypotonic environment, the extracellular solute concentration is low. There is then a
tendency for water molecules to move into the cell from outside (this, tendency is the osmotic pressure),
which causes the intracellular hydrostatic pressure to increase. (B) In an isotonic environment, the
osmotic pressure is balanced by the hydrostatic pressure, and, consequently, there is no net flow of water
molecules into or out of the cell. (C) In hypertonic environments, the extracellular solute concentration is
high, and water flows out of the cell, decreasing the internal hydrostatic pressure. As a result, the cell
shrinks.
35
(A)
(B)
Figure 3.2. (A) The replication rate of E. coli MG1655 increases approximately linearly with
temperature. (B) Average cell length (blue diamonds) and width (red circles) as a function of the
replication rate. The average cell length increases with the replication rate, whereas, the width shows a
slight decrease. Error bars show the standard error of the mean (SEM).
36
A
B
C
D
E
Figure 3.3. Representative images of E. coli MG1655 cells before (A,C) and after (B,D) the addition of
500 mM sucrose. Panels A,B show a cell grown at 23°C, and panels C,D are a cell grown at 42°C. (E)
The fractional change in length (blue diamonds) and width (red circles) increase with replication rate.
The fractional change in length varies approximately linearly, as is predicted by the mechanochemical
model. The fractional change in width is always smaller (by approximately a factor of 2) than the
fractional change in length and does not show a linear increase. Error bars denote the SEMs of the
measurements.
37
Chapter 4
The effects of peptidoglycan structure on the elasticity of the
Gram-negative cell wall
4.1 Introduction
The peptidoglycan of bacterial cells is an elastic macromolecular material which serves as the
stress bearing structure of the cell and provides the mechanical framework that defines the
bacterium’s shape. Although this structure provides rigidity for the cells, the elastic properties of
the PG also allow the cell to expand or contract under varying environmental conditions.
Furthermore, the specific elastic properties of the cell may be very important in determining
cellular morphology, as applied force has been shown to affect cell shape and how force is
distributed through the cell wall depends on the elastic properties of the material. Despite the fact
that many of the structural details of the PG are known, how PG elasticity depends on factors
such as glycan strand length and crosslink density is completely unknown.
Gram-negative cell walls are monolayered structures comprised predominantly of a repeated
chain of sugars (glycan strands) crosslinked by peptide stems forming a covalently linked
meshwork. This layer wraps around the entire circumference of the cell and spans the entire cell
length, forming roughly hemispherical caps at the cell end. The peptide chains that are created by
binding peptide stems on neighboring glycan strands are believed to be more flexible than the
relatively stiff glycan strands [19, 33]; however, the evidence that supports this belief is largely
circumstantial. Early EM data suggested that the glycan strands were arranged circumferentially,
38
with the peptide stems oriented along the long axis of the rod-shaped cell [12, 26, 70-72]. In
addition, AFM experiments suggest that the elastic modulus of the cell wall along the long axis
is between a factor of 2 to 3 less than the circumferential modulus [26], and hyperosmotic shock
experiments show a roughly two fold difference between the fractional changes in length and
width (See Chapter 3). Combining these two observations then suggests that the peptides are 2-3
times less stiff than the glycan strands, which would also imply that the PG layer is an
anisotropic elastic material. In this chapter we investigate the elasticity of a single layer of PG as
a function of the glycan strand length and the degree of crosslinking. Reversed HPLC has been
used frequently to measure glycan strand lengths giving values of 25-35 disaccharide units for
the mean glycan chain length (GCL) in exponentially growing E. coli cells [12, 27]. The mean
glycan chain length varies also for different bacterial species; ranging from 50-250 disaccharide
units in Bacilli (B. subtilis, B. licheniformis and B. cereus [73-75], about 18 disaccharide units in
Staphylococcus aureas (S. aureas) [73] and less than 10 disaccharide units for H. pylori [76]. It
would seem reasonable that the longer the glycan strands, the stiffer the elasticity of the cell wall,
which presumably would be favorable for the bacterium. However, there is evidence that glycan
strands are synthesized and inserted with lengths longer than the GCL and are then cut
enzymatically to shorter lengths. Taking this together with the fact that different species of
bacteria have different GCLs might suggest that glycan strand length is a factor that is regulated
for some purpose. However, it is not clear what that purpose is nor even how changing GCL
affects the elasticity of the cell wall. In addition, the peptide stems are of a fixed length, but not
all of these stems are crosslinked to neighboring glycan strands. The degree of crosslinking of E.
coli has been measured and varies from 40% to 60% [35, 53]. Due to the location of the peptide
stems on the glycan strands, the number of possible peptide stems is a function of glycan chain
39
length (GCL). Lower degrees of crosslinking presumably lead to weaker cell walls. Therefore,
it is expected that the Young’s modulus of the cell wall is lower in cells with lower degrees of
crosslinking.
Why, then, does bacterial cell wall synthesis seem to regulate GCL and the degree of
crosslinking? Here we make the assumption that these features are important for determining the
elastic properties of the cell wall. It is likely that the elastic properties of the PG are crucial for
creating and maintaining the shape of the cell. Therefore, determining how GCL and the degree
of crosslinking affect the elasticity of the PG is necessary for understanding bacterial
morphology. To address this question, we simulate the PG network and determine the strain that
is induced by a range of stresses. Our simulations estimate the physiological range of GCL and
degree of crosslinking that are necessary to prevent cell lysis and also suggest that these
parameters may be tuned to optimize the anisotropy of the bacterial cell wall.
4.2 Modeling the Gram-negative cell wall
The disaccharide units that build the peptidoglycan structure of the cell wall are around 1.03 nm
in length and the stretched peptide chains are on order of 4 nm [28, 77 -81]. Therefore, the
effective area of the subunit is around 5 nm2. The total surface area of E. coli or B. subtilis is
around 5 - 10 m2. A single layered cell wall for E. coli requires roughly 106 disaccharides and
the multi-layered cell wall of B. subtilis requires at least ten times more subunits than E.coli.
While computational simulations can be performed to simulate these structures [82], it is
computationally expensive and difficult to determine how changes in the physical structure
and/or biochemistry affect the gross behavior of the cell wall.
40
To overcome this large discrepancy in length scales, we constructed a discrete model of a patch
of peptidoglycan that is on the order of 0.01 m2 in size (a patch of 50x50 disaccharide units).
Each disaccharide was treated as a particle that can bind to other disaccharides through Hookean
spring-like interactions that represent either the glycosidic or the peptide bonds (Figure 4.1). The
spring stiffnesses were estimated from mechanical measurements of the Young’s modulus of
bacteria, which have been carried out in E. coli [59], B. subtilis [83], and other bacteria [83-84].
In addition, we used that the glycan strands are roughly three times stiffer than the peptide chains
[32, 26, 19, 33]. In essence, this model is conceptually similar to the computational model used
by Huang and coworkers [59, 81, 84]; however, we use realistic sizes for the disaccharides and
use our simulations to extract continuum-level parameters, such as the Young’s moduli along the
longitudinal and circumferential directions.
In order to construct the PG structure for a single layer of a patch of cell wall, which was then
used to determine the elastic moduli as a function of the GCL and degree of crosslinking, we
used a percolation type analysis, such as has been used to study many problems in randomly
connected networks [87]. We considered a 50x50 rectangular array of nodes. Each node
represents a potential disaccharide unit. In principle, any disaccharide unit can be connected to
any of its nearest neighbors in the y-direction. This direction represents the orientation of the
glycan strands, and a connection between nearest neighbors in the y-direction represent
glycosidic bonds. We define a probability Pg that sets the probability that any disaccharide unit
is connected to the neighbor above it. For each node, we use a random number generator to
select a number between 0 and 1. If the number is less than or equal to the probability, then we
connect the disaccharide to its upward neighbor. Otherwise, those nodes are not connected. A
41
similar procedure is used to define peptide crosslinks that connect the strands along the xdirection. For the peptides, we define a separate probability Pp that defines the probability that a
crosslink is formed. However, as mentioned in previous chapters, the peptide stems rotate by
approximately 90 degrees per disaccharide unit [59]. Therefore, only every 4th disaccharide can
be connected by a peptide crosslink to its neighbor to the right. For every 4th node along a given
column, we generate another random number. If this number is less than or equal to Pp, then we
connect the node to its rightward neighbor. These connections then define a randomly generated
network that has a GCL that is equal to the inverse of (1 – Pg) and a degree of crosslinking  =
Pp. The network is only a viable model for the cell wall if there are paths that connect at least one
node on each edge to at least one node on each of the other edges. In the language of percolation
theory, this represents an infinite cluster [87]. If an infinite cluster does not exist, then the cell
wall will lyse for any applied force. Therefore, we only consider networks that include an
infinite cluster. In addition, any nodes that are not connected back to the infinite cluster are
removed from the network. Two representative networks are shown in Figure 4.1.
To understand the effect of force on the material properties of these simulated PG networks, we
treat the connections as Hookean springs with spring constants that were determined as described
above.
Because the disaccharides are molecular-sized and are surrounded by fluid, we assume that the
movement of the disaccharides obeys overdamped dynamics, with a drag coefficient ; i.e., if the
position of a node is defined as X, then the time rate of change of the position obeys dX/dt) =
F, where F is the sum of the forces that act on that node and we only consider motion in the x-y
plane.We started our simulations from the unstretched state and then applied a fixed constant
42
force directed along the positive x-direction to each node on the right side of the array, and an
equal and opposite force to the nodes on the left side of the array. Likewise, we applied a force
directed along the positive y-direction to the nodes on the top edge of the array and an equal and
opposite force to the bottom edge nodes. We then solve for the dynamics of each node and run
our simulations until the network reaches equilibrium.
Using the equilibrium conformation, we then calculated the stress and the strain. The xx
component of the stress tensor was calculated as xx = Fx/Lg, where Fx was the total force that
we applied to the right side of the network, and Lg was the projected length along the y-axis of
the right side of the stretched network. The yy component was found in a similar manner using
the total force applied to the top of the network and the projected length along the x-axis of the
top side of the stretched network.
In order to calculate the strain, we first computed the displacement vector for each node, u,
which is the difference between the stretched position of the node and its initial position. The
strain tensor is related to derivatives of the displacement vector. For example, the xx component
of the strain tensor depends on the derivative of the x-component of the displacement with
respect to the x-direction. To compute this derivative using our networks, we took the average
value of the x displacement of the nodes on the right edge and subtracted them from the average
value of the x-component of the displacements for the nodes on the left edge. The difference
between these two averages was then divided by the unstretched distance between those two
sides. The resulting quantity is defined as uxx. In a similar way, we calculate the other
derivatives, such as the derivative of the x-component of the displacement along the y direction,
43
uxy, which is calculated using the difference between the average value of the x-component of the
displacement along the top edge and the average value of the x-component of the displacement
along the bottom edge divided by the unstretched length of the top edge. The strain tensor is
then given by
ij 
1
 uij  u ji  uilu jl 
2
(4.1)
4.3 Results and Discussion
In order to determine how the elastic properties of the bacterial cell wall depend on GCL and the
degree of crosslinking, we simulated random networks with a range of different GCLs and
degrees of crosslinking (as described in Sec. 4.2). We independently varied Pp (which is
equivalent to the degree of crosslinking) and Pg (which is a probability that is related to the GCL)
from 0 to 1 in increments 0.05. For each combination of Pp and Pg, we constructed ten networks,
and, for each network, we applied a range forces to the edges in the x- and y-directions, as
described in Sec. 4.2. The nodes were then moved according to the overdamped dynamic
equation and the equilibrium stress and strain was computed for every force combination. All
simulations were carried out in MATLAB.
As is well known from percolation theory [87], only certain values of the probabilities will lead
to networks that contain an infinite cluster. The consequence of this for the bacterial cell wall is
that the PG will only be a single connected macromolecule if the GCL and degree of crosslinking
are sufficiently large. The networks that we simulate reflect the structure that is commonly
44
assumed for the PG. We find that we only get infinite clusters when Pg > 0.6 and Pp > 0.25.
These values correspond to a minimum GCL of 3 and a degree of crosslinking equal to 12%. In
addition, though, the value of Pg at which the percolation transition occurs depends on Pp.
Therefore, it is not possible to have a GCL of 3 and a degree of crosslinking equal to 12%. The
white region in Figure 4.2 shows the values for 1/Ng and  = Pp where infinite clusters are not
observed (Here Ng is the average number of disaccharides per glycan strand (i.e., the GCL),
which is also equal to 1/(1-Pg)). This region represents combinations of GCL and  that should
be inaccessible to bacteria with a single-layered PG and shows that smaller values of GCL
require larger degrees of crosslinking, which is consistent with the observations that E. coli has
GCLs in the range of 25-30 and a degree of crosslinking around 20%, whereas S. aureus has a
GCL of 3-10 and a degree of 93% [31].
For networks that produced infinite clusters, we then determined the relationship between the
stress and strain. By deforming the networks using various forces applied to the edges (as
described in Sec. 4.2), we computed the diagonal components of the stress tensor as a function of
the components of the strain tensor. In general we find that both of these components of the
stress depend on the xx and yy components of the strain; however, they were not strongly
dependent on the xy component of the strain. It is likely that the xy dependence of the strain was
not important since we only applied expansive forces in our simulations and did not apply any
shear forces. We chose to only use expansive forces, because we were most interested in the
response of the cell wall to the turgor pressures. We found that both components of the stress
depended nonlinearly on both diagonal components of the strain. Therefore, we fit the stress vs.
strain curves to the following quadratic functions:
45
(4.2)
The Young’s modulus of a material defines the linear relationship between a given component of
the stress and the same component of the strain. When the stress depends nonlinearly on the
strain, the Young’s modulus, therefore, depends on the strain. A general expression for the
Young’s modulus is then that the modulus along a given direction Eij = dij/dij. In order to
obtain a single value for the Young’s moduli of the PG in the longitudinal (peptide) and
circumferential (glycan) directions, Ep and Eg, repectively, we used the expressions given in Eq.
4.2 to define the Young’s moduli around a strain of 0.2 (which is approximately the strain that is
observed in our hyperosmotic shock experiments (Chapter 3).
Using this formalism, we found that the longitudinal elastic modulus (i.e, the modulus along the
peptide bonds) decreases with decreases in the GCL and the degree of crosslinking (Figure
4.2A). This result is somewhat expected as it states that as the PG network is degraded it should
get softer. The dependence of the Young’s modulus along the circumferential direction on the
GCL and degree of crosslinking was more surprising. While, in general, decreasing either of
these factors lead to smaller values of Eg, we found that for large value of the GCL (for glycan
binding probabilities near 1, Pg > 0.95) that the maximum value of Eg was located near  = 0.7
(Figure 4.2B). This implies that when the GCL is greater than 20 disaccharides, decreasing the
degree of crosslinking from 1 to 0.7 leads to increases in the stiffness of the cell wall in the
circumferential direction. Because the longitudinal stiffness decreases in the same regime, this
46
suggests that the anisotropy of the cell wall is optimized for values of  = 0.7 (which
corresponds to a degree of crosslinking around 35%).. Since E. coli has a GCL of 25-35
disaccharides and a degree of crosslinking around30%, this suggests that the biochemical
kinetics of synthesis may be regulated to optimize the anisotropy of the cell wall. It may, then,
be that this anisotropy is important for maintaining the width of the rod-shaped cell. However,
the work presented here only suggests this possibility. More work is needed to validate this
hypothesis.
47
Figures
(A)
(B)
Figure 4.1. Peptidoglycan percolation models for a gram-negative bacterium at (A) 100%
connectivity (i.e., Pp = Pg = 1), and (B) 50% crosslink density (Pp = 0.5) and 96% mean glycan chain
length (Pg = 0.96), which is within the range appropriate for E. coli [10, 32, 18, 64]. (Red) represents
glycosidic bonds and black represents peptide bonds.
48
(A)
(B)
Figure 4.2. Elastic moduli of a single layer of peptidoglycan at varying crosslink densities and glycan
chain length. (A).Elastic modulus, Ep decreases with either a decrease in degree of crosslinkage or a
decrease in glycan chain length. Ep increases with an increase in peptide crosslinkage, Ψ for large values
of Pg. (Dotted lines) (B) Elastic modulus, Eg shows peak at intermediate values of peptide crosslinkage, Ψ
for longer glycan strands. Eg increases when peptide crosslinkage, Ψ is around 0.7 for large values of Pg.
(Dotted lines)
49
Chapter 5
Conclusions and suggestions for future research
At face value, the ability of a rod-shaped bacterial cell to grow in length without changing its
width does not seem that difficult or interesting. The bacterial cell is just like a balloon, where
the elastic cell wall material is analogous to the latex and is inflated by the turgor pressure. For
many years, the scientific community held this belief. Then, in the early part of this century, the
discovery that bacteria possess cytoskeletal proteins structurally homologous to actin that are
necessary for proper cell shape led to the realization that the creation and regulation of bacterial
morphology is much more complex than we had imagined. Research over the past 15 years has
now clearly shown that we still don’t understand much about how even a simple rod-shaped
bacterium controls its width. In addition, we now also know that bacterial morphology can be
influenced by external forces.
The work presented in this dissertation sought to begin to develop a more mechanistic picture of
bacterial cell wall synthesis. We took the hypothesis that bacterial morphology is a consequence
of the interplay between the biochemical kinetics of synthesis and the biophysical interactions
due to the material properties of the PG and the expansive force from the turgor pressure.
Working from this hypothesis, we started simple and aimed to understand the long standing
observation that E. coli cells that replicate faster are also longer. While this observation is well
known and has been reported by many groups, no explanation for why it happens has been
proposed. To investigate the mechanisms that could lead to this phenomenon, we considered
three of the basic biochemical processes involved in cell wall synthesis: severing of existing
crosslinks, insertion of new glycan strands, and re-crosslinking of the PG. We simplified our
50
analysis by considering a simple structure for the PG where a fixed number of glycan monomers
are arranged circumferentially. We then developed a model based on the biochemical kinetics of
severing and crosslinking of the peptides. We included in this model the biophysics of the
stretching of the cell wall by the turgor pressure. We found that this simple model could only
reproduce the observation that faster replicating cells are longer if the severing and insertion
rates were proportional to the replication rate and the crosslinking rate was independent of the
replication rate. Comparing the model to the observed dependence of the length on replication
rate allowed us to predict values of the biochemical kinetic rates and turgor pressure inside the
cells. In addition, the model predicts that the PG in faster growing cells is weaker and therefore
should be stretched more by the turgor pressure.
Next, we tested the predictions of our mechanochemical model by using hyperosmotic shock to
effectively reduce the turgor pressure inside the cells. We shocked E. coli cells that were grown
at different temperatures and, therefore, had different replication rates. We found that the
fractional change in length was linearly proportional to the replication rate, as predicted by the
model. Interestingly, without changing the parameters in our model from what we estimated
from the dependence of length on replication rate, we were able to accurately predict the
dependence of the fractional change in length on the replication rate. These results provide good
support for our mechanochemical model.
While our experiments provide credence to our mechanochemical model, the model itself is too
simple to address larger questions in bacterial morphology. Our assumption that the number of
glycan monomers about the circumference of the cell is fixed does not allow us to investigate the
biochemical mechanisms involved with glycan strand elongation and glycan chain severing,
which are likely important for determining bacterial morphology.
51
To investigate this further, we constructed a discrete model of a growing patch of the bacterial
cell wall, where we were able to vary the average number of disaccharides per glycan strand and
the fraction of crosslinked peptides. We constructed random networks using a percolation type
analysis and represented glycosidic bonds and peptide chains as Hookean springs. Our model
predicts the values of GCL and degree of crosslinking that are required to create a single
macromolecular PG structure, which explains why bacterial cells with shorter GCL are typically
more crosslinked. Then by applying a series of forces to the edges of this simulated PG patch,
we were able to compute the dependence of the effective Young’s moduli as a function of glycan
chain length and the degree of crosslinking. We found that the Young’s modulus along the
longitudinal axis of the cell should decrease with decreases in GCL or degree of crosslinking.
Interestingly, though, we found that the circumferential Young’s modulus has a peak at around
35% peptide crosslinking when the GCL is larger than 20 disaccharides. Therefore, for PG
structures with GCLs over 20, there is a degree of crosslinking that optimizes the elastic
anisotropy of the cell wall. Since E. coli has a GCL of 25-35 disaccharides and a crosslinking
fraction of around 20-30%, it may be that these values are regulated to maintain cell wall
anisotropy.
The work presented here, therefore, lays the groundwork for a more mechanistic understanding
of bacterial morphology and will be very beneficial to the advancement of this field towards
understanding how bacteria create and maintain their shapes. Our mathematical model for cell
wall synthesis may also aid in developing novel antibiotic treatments.
Completing the discrete model by additionally simulating the insertion of newly synthesized
material would allow us to get a mode detailed outlook at the biochemical reactions in the
52
bacteria cell. Since the cell wall of Gram-positive bacteria is thicker than the Gram-negative
bacterium highlighted in our models, expanding the work done in this dissertation to account for
the structure of the cell wall in Gram-positive cells would be beneficial to understanding how the
biochemical kinetics of cell wall synthesis and the material properties of the PG differ in this
important group of bacteria.
53
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